Photovoltaics: Band Diagram

In the previous post we discussed silicon, which is the most used material in photovoltaics. In this post, we introduce the band diagram, for which we will use silicon as an example. We will start our discussion of the band diagram with the Bohr model of the silicon atom. In semiconductor materials the outer shell of the atom, which is called the valence shell, is not completely filled. The outer shell of silicon contains 4 out of the possible 8 electrons, which we call valence electrons. As we discussed in the previous post, each silicon atom in a crystalline structure is bonded to four other silicon atoms. The bonds between the silicon atoms are called covalent bonds. These bonds actually consist of two valence electrons that are shared by two silicon atoms.

All valence electrons are fixed in the lattice, forming covalent bonds, and are therefore immobile. However, at a temperature above absolute zero, thermal energy is supplied to these miconductor and some of the valence electrons are released from the covalent bonds. These excited electrons are mobile and can move around in the material, allowing the semiconductor material to conduct electricity.

The band diagram is a convenient way to express the energy state of a valence electron and is therefore a much-used tool in semiconductor physics. The y-axis in a band diagram represents the energy level of valence electrons in a semiconductor. In an isolated Si atom, electrons are allowed to have only discrete energy values. However, the periodic atomic structure of a silicon crystal results in the ranges of discrete energy states for electrons.

This band of allowed energies is known as the valence band. The valence band comprises the energy levels of all valence electrons in a semiconductor material at absolute zero. When a valence electron receives additional energy, either from a photon or through thermal vibrations, it can become mobile. Having absorbed additional energy, these mobile electrons are at a higher energy level than the bound valence electrons. The band of allowed energy states for these conduction electrons is known as the conduction band.

The lowest allowed energy level of the conduction band is known as the conduction band edge, indicated by E-C. The highest allowed energy level of the bound valence electrons is known as the valence band edge, indicated by E-V. Between the band edges is a range of energy levels that the electrons cannot occupy. This range of forbidden energy levels is known as the bandgap. The bandgap energy, indicated by E-G, is therefore equal to the difference between the conduction band edge and the valence band edge. Since in photovoltaics we are mostly interested in the energy of band edges, the levels in the bands are usually removed. The bandgap energy is a defining property of a semiconductor material. For instance, the bandgap energy of silicon at room temperature is equal to 1.12 electronvolts. So elements of which the valence shell is about half-full are semiconductors.

Elements with valence shells that are completely filled, or almost completely filled have very different electrical properties. The valence electrons of these elements are strongly bound to the atom. These elements have a very large bandgap and the valence electrons require a very large amount of energy to be released from the valence shell. As a result, these elements have a relatively small amount of mobile electrons at room temperature, and therefore conduct electricity very poorly. These elements are called insulators. Metals, on the other hand, have only 1 or 2 electrons in their valence shell, which are very loosely bound. The valence and conduction bands of metals overlap, which means that no external energy source is required for a valence electron to become mobile. Metals, therefore, have a very large amount of mobile electrons and conduct electricity very well. That is why metals are called conductors. Semiconductors have band-gap energies that are small enough for photons to excite valence electrons to the energy levels of the conduction band. This is the first step to convert energy of light directly into electricity.

We will look at the properties of the semiconductor material in the so called energy-momentum space of the electrons. On the vertical axis we have the energy state in the electronic bands. On the horizontal axis we have got the momentum of the charge carrier. This momentum is also called the crystal momentum. As discussed before, a photon can change the energy position of the electron. Just as light can be described as wave and as particle, the lattice vibrations can also be described as waves and as particles, which we call phonons. A phonon can change the momentum of an electron. Important to realize is that the position of the valence and conduction band may differ indifferent direction of the lattice coordination. For indirect band gap the highest point of the valence band and the lowest point of the conduction band are not aligned. This means that exciting an electron from the valence to conduction band requires energy provided by a photon and momentum, provided by a phonon. In contrast, in the energy-momentum space a direct band gap has the highest point of the valence band vertically aligned with the lowest point of the conduction band. Exciting an electron from the valence to conduction band, therefore, requires only the energy provided by a photon without any additional momentum transfer.

This means that the excitation of an electron induced by photon absorption is more likely to happen in direct band-gap materials than in indirect band-gap materials. For direct band-gap materials no additional momentum matching coming from phonons is required. As such, the absorption for direct band-gap materials is significantly higher than for indirect band gap materials. For example, the important PV materials are silicon and Gallium Arsenide. Silicon is an indirect band-gap material whereas GaAs is a direct band-gap material. As we will discuss later in this course, for the same thickness of material, GaAs will absorb more light in the visible spectrum than silicon will do.